\typeout{Abstract}
\setcounter{page}{6}
%\addcontentsline{toc}{chapter}{Abstract}
\begin{center}
\Large{\rm\textbf{Abstract}}
\end{center}
\vspace{4ex}

The Fourier transform, the quantum mechanical harmonic oscillator, and supersymmetric quantum mechanics are well-studied objects in mathematics. The relations between them are also well-understood. The traditional understanding of each appears to suggest rigid structures that do not leave much flexibility for alternatives. In this thesis, we will develop generalizations of the Fourier transform and explore their relations. Exploring the analytic structure of these integral transforms, particularly their related Hamiltonians, naturally leads to Hamiltonian-like operators which have analytic and algebraic structures.

The algebraic structure underlying the Hamiltonian-like operators associated to generalized Fourier transforms suggests a much-more-general abstract formulation. To this end, we introduce the coupled supersymmetry (coupled SUSY) algebraic framework which unifies the quantum mechanical harmonic oscillator and supersymmetric quantum mechanics in a more complete way. The coupled SUSY framework subsumes the quantum mechanical harmonic oscillator and provides a broader class of systems which have interesting functional analytic and algebraic structures. In this setting, one is able to develop further generalizations of the Fourier transform. A further generalization of coupled SUSY is briefly presented which appears to offer new insights into supersymmetry.


\endinput